American Institute of Mathematical Sciences

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June  2012, 17(4): 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

Asymptotics of blowup solutions for the aggregation equation

Yanghong Huang 1, and  Andrea Bertozzi 2,

1. 

Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada
2. 

520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095

Received  January 2011 Revised  September 2011 Published  February 2012

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.
Keywords: asymptotic behavior, self-similar solutions., blowup, Aggregation equation.
Mathematics Subject Classification: Primary: 35B40, 35B44; Secondary: 92D5.
Citation: Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008).   Google Scholar

[2]

D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion,, J. Nonlinear Sci., 5 (1995), 29.   Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[4]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation,, Commun. Math. Sci., 8 (2010), 45.   Google Scholar

[5]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683.  doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[6]

A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,, to appear in SIAM J. Math. Anal., (2012).   Google Scholar

[7]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$,, Comm. Math. Phys., 274 (2007), 717.  doi: 10.1007/s00220-007-0288-1.  Google Scholar

[8]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pur. Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[9]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models,, J. Differential Equations, 222 (2006), 341.  doi: 10.1016/j.jde.2005.07.025.  Google Scholar

[10]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939.  doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar

[11]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion,, Networks and Heterogeneous Media, 3 (2008), 749.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[12]

E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows,, Arch. Ration. Mech. Anal., 163 (2002), 329.  doi: 10.1007/s002050200204.  Google Scholar

[13]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations,, Duke Math. J., 156 (2011), 229.  doi: 10.1215/00127094-2010-211.  Google Scholar

[14]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system,, Phys. D, 232 (2007), 33.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[15]

H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions,, Communications in Mathematical Physics, 304 (2011), 649.  doi: 10.1007/s00220-011-1237-6.  Google Scholar

[16]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).   Google Scholar

[17]

J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations,, Nonlinearity, 22 (2009).  doi: 10.1088/0951-7715/22/1/R01.  Google Scholar

[18]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Phys. Rev. Lett., 95 (2005).  doi: 10.1103/PhysRevLett.95.226106.  Google Scholar

[19]

D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles,, Phys. D, 220 (2006), 183.  doi: 10.1016/j.physd.2006.07.010.  Google Scholar

[20]

Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'', Ph.D thesis, (2010), 10.   Google Scholar

[21]

Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$,, SIAM J. Appl. Math., 70 (2010), 2582.  doi: 10.1137/090774495.  Google Scholar

[22]

Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions,, preprint, (2010).   Google Scholar

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941.   Google Scholar

[25]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[26]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.  doi: 10.1007/s002850050158.  Google Scholar

[27]

A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation,, J. Math. Biol., 47 (2003), 353.  doi: 10.1007/s00285-003-0209-7.  Google Scholar

[28]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49.  doi: 10.1007/s00285-004-0279-1.  Google Scholar

[29]

C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts,, The European Physical Journal-Special Topics, 157 (2008), 93.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[30]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[31]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[32]

G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

[33]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58,, American Mathematical Society, (2003).   Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008).   Google Scholar

[2]

D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion,, J. Nonlinear Sci., 5 (1995), 29.   Google Scholar

[3]

D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media,, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615.   Google Scholar

[4]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation,, Commun. Math. Sci., 8 (2010), 45.   Google Scholar

[5]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683.  doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[6]

A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,, to appear in SIAM J. Math. Anal., (2012).   Google Scholar

[7]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$,, Comm. Math. Phys., 274 (2007), 717.  doi: 10.1007/s00220-007-0288-1.  Google Scholar

[8]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation,, Comm. Pur. Appl. Math., 64 (2011), 45.  doi: 10.1002/cpa.20334.  Google Scholar

[9]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models,, J. Differential Equations, 222 (2006), 341.  doi: 10.1016/j.jde.2005.07.025.  Google Scholar

[10]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939.  doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar

[11]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion,, Networks and Heterogeneous Media, 3 (2008), 749.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[12]

E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows,, Arch. Ration. Mech. Anal., 163 (2002), 329.  doi: 10.1007/s002050200204.  Google Scholar

[13]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations,, Duke Math. J., 156 (2011), 229.  doi: 10.1215/00127094-2010-211.  Google Scholar

[14]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system,, Phys. D, 232 (2007), 33.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[15]

H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions,, Communications in Mathematical Physics, 304 (2011), 649.  doi: 10.1007/s00220-011-1237-6.  Google Scholar

[16]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006).   Google Scholar

[17]

J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations,, Nonlinearity, 22 (2009).  doi: 10.1088/0951-7715/22/1/R01.  Google Scholar

[18]

D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Phys. Rev. Lett., 95 (2005).  doi: 10.1103/PhysRevLett.95.226106.  Google Scholar

[19]

D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles,, Phys. D, 220 (2006), 183.  doi: 10.1016/j.physd.2006.07.010.  Google Scholar

[20]

Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'', Ph.D thesis, (2010), 10.   Google Scholar

[21]

Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$,, SIAM J. Appl. Math., 70 (2010), 2582.  doi: 10.1137/090774495.  Google Scholar

[22]

Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions,, preprint, (2010).   Google Scholar

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

T. Laurent, Local and global existence for an aggregation equation,, Comm. Partial Differential Equations, 32 (2007), 1941.   Google Scholar

[25]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows,, Arch. Ration. Mech. Anal., 172 (2004), 407.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[26]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534.  doi: 10.1007/s002850050158.  Google Scholar

[27]

A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation,, J. Math. Biol., 47 (2003), 353.  doi: 10.1007/s00285-003-0209-7.  Google Scholar

[28]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49.  doi: 10.1007/s00285-004-0279-1.  Google Scholar

[29]

C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts,, The European Physical Journal-Special Topics, 157 (2008), 93.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[30]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups,, SIAM J. Appl. Math., 65 (2004), 152.  doi: 10.1137/S0036139903437424.  Google Scholar

[31]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[32]

G. Toscani, One-dimensional kinetic models of granular flows,, M2AN Math. Model. Numer. Anal., 34 (2000), 1277.  doi: 10.1051/m2an:2000127.  Google Scholar

[33]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58,, American Mathematical Society, (2003).   Google Scholar

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